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In the number partitioning problem (NPP) one aims to partition a given set of $N$ real numbers into two subsets with approximately equal sum. The NPP is a well-studied optimization problem and is famous for possessing a…
Given two elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit an isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest…
We develop the first quantum algorithm for the constrained portfolio optimization problem. The algorithm has running time $\widetilde{O} \left( n\sqrt{r} \frac{\zeta \kappa}{\delta^2} \log \left(1/\epsilon\right) \right)$, where $r$ is the…
Let $E$ be an elliptic curve over the finite field $\mathbb F_q$. We prove that, when $n$ is a sufficiently large positive integer, $\#E(\mathbb F_{q^n})$ has a prime factor exceeding $n\exp(c\log n/\log\log n)$.
The {\em maximum cardinality} and {\em maximum weight matching} problems can be solved in time $\tilde{O}(m\sqrt{n})$, a bound that has resisted improvement despite decades of research. (Here $m$ and $n$ are the number of edges and…
Approximate proof labeling schemes were introduced by \\Censor-Hillel, Paz and Perry \cite{CPP}. Roughly speaking, a graph property~$\cP$ can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph…
The emergence of large language models (LLMs) like GPT-4 has revolutionized natural language processing (NLP), enabling diverse, complex tasks. However, extensive token counts lead to high computational and financial burdens. To address…
The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, however, we still have a far more limited understanding of maximal matching which is one…
The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach,…
We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is $C^1$ in the velocity. Such an ODE arises as a model of spin-orbit coupling in a star/planet system, and the motivation…
Significance testing -- especially the paired-permutation test -- has played a vital role in developing NLP systems to provide confidence that the difference in performance between two systems (i.e., the test statistic) is not due to luck.…
Persistent homology is a crucial invariant that is used in many areas to understand data. The $O(N^4)$ run time is a hindrance to its use on most large datasets. We give a parallelization method to utilize multi-core machines and clusters.…
Let $f$ be a fixed (holomorphic or Maass) modular cusp form. Let $\cq$ be a Dirichlet character mod $q$. We describe a fast algorithm that computes the value $L(1/2,f\times\chi_q)$ up to any specified precision. In the case when $q$ is…
We present an elementary proof of the group properties of the elliptic curve known as "Curve25519", as a component of a comprehensive proof of correctness of a hardware implementation of the associated Diffie-Hellman key agreement…
We investigate the practical aspects of computing the necessary and possible winners in elections over incomplete voter preferences. In the case of the necessary winners, we show how to implement and accelerate the polynomial-time algorithm…
We present a new on-line algorithm for computing the Lempel-Ziv factorization of a string that runs in $O(N\log N)$ time and uses only $O(N\log\sigma)$ bits of working space, where $N$ is the length of the string and $\sigma$ is the size of…
The stabiliser formalism plays a central role in quantum computing, error correction, and fault tolerance. Conversions between and verifications of different specifications of stabiliser states and Clifford gates are important components of…
Advanced embedded algorithms are growing in complexity and they are an essential contributor to the growth of autonomy in many areas. However, the promise held by these algorithms cannot be kept without proper attention to the considerably…
Computing the simulation preorder of a given Kripke structure (i.e., a directed graph with $n$ labeled vertices) has crucial applications in model checking of temporal logic. It amounts to solving a specific two-players reachability game,…
We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given…